14) Polar Coordinates

Butterflies

Polar equations of the form r =1+sin(nTheta)+(cos2ntheta)^2

> plot([1+sin(2*theta)+(cos(4*theta))^2,theta,theta =0..2*Pi], coords=polar);

> plot([1+sin(3*theta)+(cos(6*theta))^2,theta,theta =0..2*Pi], coords=polar);

Arc Length in Polar Coordinates

If r = f(theta), the arc length from theta =alpha to theta =beta is given by

> restart;

> LP:=Int (sqrt(f(theta)^2+D(f)(theta)^2) ,theta=alpha..beta);

Find the arc length of r=1+cos (theta) without using symmetry.

> f:=(theta)->1+cos(theta);

> plot([f(theta),theta,theta=0..2*Pi], coords=polar);

> Int (sqrt(f(theta)^2+D(f)(theta)^2) ,theta=0..2*Pi);

> simplify(%);

> value(%);

Do the same problem using symmetry

> 2*int (sqrt(f(theta)^2+D(f)(theta)^2) ,theta=0..Pi);